Moduli space theory for complete, constant Q-curvature metrics on finitely punctured spheres

Abstract

We study constant Q-curvature metrics conformal to the round metric on the sphere with finitely many point singularities. We show that the moduli space of solutions with finitely many punctures in fixed positions, equipped with the Gromov-Hausdorff topology, has the local structure of a real analytic variety with formal dimension equal to the number of the punctures. If a nondegeneracy hypothesis holds, we show that a neighborhood in the moduli space is actually a real-analytic manifold of the expected dimension. We also construct a geometrically natural set of parameters, construct a symplectic structure on this parameter space and show that in the smooth case a small neighborhood of the moduli space embeds as a Lagrangian submanifold in the parameter space.